I am building a portable trolley beam arrangement to move large stones
in building a staircase. Most of the stones are around 1,500 lbs.
Maximum load is 2,000 lbs which can, with the trolley, be in any
position on the long beam. The trolley beam has a span of 13'. It
rests on the center of another beam at each end (at right angles)which
each have a span of 6'. The beams form an "I" in plan. I want to
calculate the size of the beams required.  I already own a 14' long
6x12 beam (4" wide, flat flanges} which I hope may be strong enough
for the long beam, but if not, I could cut it for the two short beams,
but because this is intended to be portable, I want to keep it as
light as possible. I do have the option of adding another support to
shorten the span if a particular stone is too heavy for my beams.
If the 6x12 is two small for 2,000 lbs, can you tell me its capacity
as a trolley beam?
I would like to know what the safety factor is in such calculations
and if the stone mason misjudges the load, and exceeds the "safe"
limit, how much overloading is, in practical terms, expected before
deforming the beams? (I want to be sure the stone mason is well
informed.)
In short:
What beams should I use to move up to 2,000 lbs over a 13'span
supported by 6' spans?
How much weight could I support on a trolley with my 6x12, on a 13' span?
Generally speaking, what percentage of load is required beyond the
safe limits to deform the beams?
Thank you,
Joseph

Hello josephth, I think we can come up with a safe design for you.
Safety factor for designs involving lifting of heavy loads are
generally taken to be the ultimate strength of the material divided by
six. A structural beam normally is considered to have an ultimate
strength (Su) of 50,000 psi. This will give us a design stress of:

Sd = Su/6 = 8,300 psi

The reasons for such a large safety factor are that design loads are
often exceeded, devices may be damaged, and large shock loads may
occur due to rigging slips or failure.

In designing a beam either the bending moment or the allowable
deflection rules. Generally in short beams (such as both of yours)
deflection is not a concern. I checked deflection just to make sure
and it does not affect beam selection in your design. So, I will not
include those calcs just to make things as simple as possible.

6' beam design:

We can take the worst case where the trolley is adjacent to one of the
end beams and assume that the 2,000# is fully applied at the center of
the beam. The maximum bending moment (M) is:

M = P x l/4 = 2,000# x 72 in. / 4 = 36,000 in-lb

The required section modulus (S) is:

S = M/Sd = 36,000 / 8,300 = 4.33 in^3

13' beam design:

The worst case here is when the trolley is at the center of the beam. So, we have:

M = Pl/4 = 2,000# x 156 in. / 4 = 78,000 in-lb

S = M/Sd = 78,000 / 8,300 = 9.39 in^3

The section modulus of your W6x12 beam is 7.25 in^3. It would be fine
for the 6' beams. The 13' span beam requires a beam with a section
modulus at least  9.39 in^3. A good choice would be a W6x16 with a
value of 10.2 in^3.

If you have any questions after reading this, please ask for a
clarification and I will be glad to explain further.

Good luck with your stone work, Redhoss